Thick Plate Problems Research Articles

Thermal buckling analysis of porous FGM plates

The interest of this study is the analysis of the thermal buckling problem of porous thick rectangular plates made of FGM material using high order shear theory. Material properties like Young's modulus and coefficient of thermal expansion of the plate are assumed to vary continuously in the thickness direction as a function of the volume fraction of the constituents defined with the modified mixture rule including the volume fraction of porosity with three different types of porosity distribution.The equilibrium and stability equations are obtained based on the present theory. The nonlinear governing equations are solved for simply supported plates. The thermal loads are assumed to be uniform, linear and non-linear distribution across the thickness. A parametric study will be carried out in order to underline the effects of the various parameters governing the response of thick FGM plates on the critical buckling temperature.

New benchmark free vibration solutions of non-Lévy-type thick rectangular plates based on third-order shear deformation theory

Higher-order shear deformation theories are generally adopted to study the dynamic behavior of substantially thick plates, of which Reddy’s third-order deformation theory (TSDT) is most extensively used in view of its conciseness and high accuracy. However, due to the intractability in solving the governing higher-order partial differential equations, analytic solutions of TSDT-based thick plate problems are limited to very few special cases, i.e., Navier-type or Lévy-type solutions. Since analytic solutions can not only serve as benchmarks for validation of various methods, but are also useful for structural designs, especially at the early design stage, developing new analytic methods that are applicable to more common non-Lévy-type thick plates is of much importance. In this paper, for the first time, a novel symplectic superposition method, which was proposed originally for thin/moderately thick plate problems, is extended to free vibration problems of thick rectangular plates based on the TSDT. Besides yielding Lévy-type solutions, new non-Lévy-type analytic free vibration solutions are obtained. Comprehensive benchmark natural frequencies and mode shapes are tabulated/plotted, all of which are well validated by the 3D theory-based solutions from the finite element method (FEM) and the literature, if any. The present method is implemented in the symplectic space under the Hamiltonian system framework, rather than in the Euclidean space under the Lagrangian system framework as implemented in conventional methods, providing a new route to more analytic solutions for complex plate and shell problems that have not been explored due to mathematical challenge.

A boundary method using equilibrated basis functions for bending analysis of in-plane heterogeneous thick plates

A simple boundary method is developed for the solution of isotropic thick plates with in-plane arbitrarily variable material properties or thickness. Equilibrated basis functions which have proved to be effective in a variety of problems, are adopted for the bending problem of thick plates. The bases are created through a weighted residual approach over a fictitious rectangular domain so as to approximately satisfy the partial differential equation of the problem. This omits the necessity of the bases to analytically satisfy the equilibrium, thus simplifying the application of the method. Boundary conditions are applied to the approximate solution through a collocation technique, which considerably reduces its computational expenses. Mindlin’s first order and Levinson’s third-order shear deformation theories are adopted for the formulation. To accommodate more complicated geometries, a simple domain decomposition approach is also developed.

First Principles Derivation of Displacement and Stress Function for Three-Dimensional Elastostatic Problems, and Application to the Flexural Analysis of Thick Circular Plates

In this study, stress and displacement functions of the three-dimensional theory of elasticity for homogeneous isotropic bodies are derived from first principles from the differential equations of equilibrium, the generalized stress – strain laws and the geometric relations of strain and displacement. It is found that the stress and displacement functions must be biharmonic functions. The derived functions are used to solve the elasticity problem of finding stresses and displacement fields in a thick circular plate with clamped edges for the case of uniformly distributed transverse load over the plate domain. Superposition of second to sixth order Legendre polynomials which are biharmonic functions are used in the thick circular plate problem as the stress function with the unknown constants as the parameters to be determined. Use of the stresses and displacement fields derived in terms of the stress and displacement function yielded the stress fields and displacement fields in terms of the unknown constants of the biharmonic stress function. Enforcement of the boundary conditions yielded the unknown constants, leading to a complete determination of the stress and displacement function for the stress fields and the displacement fields. The solutions obtained are comparable to solutions in the technical literature.

New analytical solutions for vibration problem of thick plates

Exact solutions for the vibrations and stability problems in the mechanics of solids are sufficiently rare. For rectangular thick plates, exact solutions in the form of trigonometric series were constructed only for the case when all or two opposite sides of the plate are simply-supported. The discussion about the possibility of constructing exact solutions continues to this day. As a rule, an approximate solution is constructed in the analytical form on the basis of a variational approach. It should be noted, that as frequency increases, the number of basic functions involved in the solution also has to be increased, consequently the solutions of such type are inefficient for describing a structural element within the framework of such methods as the Continuous Element method, the Spectral Element method and the Dynamic Stiffness method. In the present research, for the first time, the analytical solutions for the vibration problem of thick orthotropic plates are obtained. The modified trigonometric basis is used to construct the general solution for the free vibration problem of the plate in a series, permitting the derivation of an infinite system of linear algebraic equations. Cases of practically important boundary conditions of completely free sides and fully clamped sides are considered. It should be noted that the presented analytical solutions for FFFF and CCCC boundary allows to completely describe the structural element in the form of a plate by means of a dynamic stiffness matrix and their use for modeling more complex structures, also in the framework of methods such as Continuous Element method, Spectral Element method and Dynamic Stiffness method. The obtained results could also be applied in projecting constructions, in developing new devices and in the optimization of their parametersin projecting of constructions, in developing new devices and in the optimization of their parameters.

Viscoelastic Behavior of Shear-Deformable Plates

The purpose of this study is to extend a new mixed-type finite element (MFE) model, developed earlier by the present authors for the analysis of viscoelastic Kirchhoff plates [Aköz, A. Y., Kadıoğlu, F. and Tekin, G. [2015] “Quasi-static and dynamic analysis of viscoelastic plates”, Mechanics of Time-Dependent Materials 19(4), 483–503], to study the quasi-static and dynamic responses of first-order shear-deformable (FSD) linear viscoelastic Mindlin–Reissner plates. In this context, various viscoelastic material models are discussed for the plate structure to read from them possible patterns of viscoelastic behavior. The developed MFE named VPLT32 is C0-continuous four-node linear isoparametric plate element with eight degrees of freedom per node. Hereditary integral form of the constitutive law with constant Poisson’s ratio is used. A new functional in the Laplace–Carson domain suitable for MFE formulation in the same domain is developed by employing Gâteaux differential (GD) method. The unique aspects of this study and the possible contributions of the proposed method to the literature can be summarized as follows: by using this new functional, moment and shear force values that are important for engineers can be obtained directly without any mathematical operation. In addition, geometric and dynamic boundary conditions can be obtained easily and a field variable can be included to the functional systematically. Moreover, shear-locking problem can be eliminated by using the GD method. Dubner and Abate numerical Laplace inversion technique is adopted to transform the obtained solution from the Laplace–Carson domain into the real-time domain. A set of numerical examples are presented not only to demonstrate the validity and accuracy of the proposed MFE formulation but also to examine the effects of load, geometry and material parameters on the viscoelastic response of FSD Mindlin–Reissner plates and to give a better insight into time-dependent behavior of engineering thick plate problems.

Natural Frequencies of Rectangular Plate With- and Without-Rotary Inertia

A nine-node isoparametric plate element, in conjunction with first-order shear deformation theory, was used for free vibration analysis of rectangular plates. Both thick and thin plate problems were solved for various aspect ratios and boundary conditions. In this work, the primary focus is on the effect of rotary inertia on the natural frequencies of rectangular plates. It is found that rotary inertia significantly affects thick plates, while it can be ignored for thin plates. The numerical convergence is very rapid and based on a comparison with data from the literature; it is proposed that the present formulation can yield highly accurate results. Finally, some numerical solutions are provided here, which may serve as benchmarks for future research on similar problems.

Free vibration analysis of rectangular plates with central cutout

AbstractA nine-node isoparametric plate element in conjunction with first-order shear deformation theory is used for free vibration analysis of rectangular plates with central cutouts. Both thick and thin plate problems are solved for various aspect ratios and boundary conditions. In this article, primary focus is given to the effect of rotary inertia on natural frequencies of perforated rectangular plates. It is found that rotary inertia has significant effect on thick plates, while for thin plates the rotary inertia term can be ignored. It is seen that the numerical convergence is very rapid and based on comparison with experimental and analytical data from literature, it is proposed that the present formulation is capable of yielding highly accurate results. Finally, some new numerical solutions are provided here, which may serve as benchmark for future research on similar problems.

Application of DKMQ element for composite plate bending structures

This paper presents an application of DKMQ plate bending element to analyze composite structures. DKMQ element is an element proposed by Katili that passed the patch test and gives good results in many plate bending applications. It can be used to analyze thick and thin plate problem without shear locking. The application of DKMQ plate bending element for composite plate has been analyzed; the convergence of displacement is presented. As validation, the solution of Srinivas and Pagano are employed. Moreover, measuring the convergence using the s-norm is addressed. DKMQ plate bending element gives a good performance and can be used as one of tools in analyzing composite structure.

SHEAR AND VOLUMETRIC LOCKING EFFECT ON THE PERFORMANCE OF HARMONIC SOLID RING FINITE ELEMENTS

Harmonic solid ring finite elements are commonly used in the analysis of axisymmetric structures subjected to non-axisymmetric as well as axisymmetric loadings. Depending on the material and/or geometrical properties of axisymmetric problems the finite element analysis may produce erroneous solutions due to approximations assumed in the formulation. Volumetric and shear locking are the some troublesome behaviors of some finite elements. In this study, finite element formulations of 4-noded (Ring4) and 9-noded (Ring9) ring elements are developed considering constant and linear shear locking effect for the element types, respectively, by incorporating selectively reduced integration technique. A computer program is coded in Matlab for the purpose and the performances of both elements are explored in terms of locking issue as well as accuracy. For this purpose several axisymmetric problems are solved such as hollow thick cylinder and circular plate problems. Numerical results indicate that while Ring9 does not suffer from volumetric locking for high values of Poisson’s ratios Ring4 suffers. Besides, while Ring4 with full integration displays shear locking effects Ring4 with selectively reduced integration eliminates the locking. The finite element formulations areexplained in detail and the results of numerical examples are presented comparatively in graphical and tabular formats

Symplectic Superposition Method for Benchmark Flexure Solutions for Rectangular Thick Plates

AbstractBenchmark flexure solutions for rectangular thick plates with combinations of clamped and simply supported edges, as well as for cantilever rectangular thick plates, are obtained by using an up-to-date symplectic superposition method in the Hamiltonian system. The method developed offers a rational way to obtain sufficiently accurate solutions to thick-plate problems, which can yield more benchmark solutions to similar problems. Comprehensive numerical results are presented for the future validation of various approximate/numerical methods.

Large increment method for elastic and elastoplastic analysis of plates

The displacement-based finite element method (FEM) has become a method of choice for material nonlinear analysis of plates. For material nonlinear problems, the displacement-based FEM relies upon a step-by-step incremental approach and the repetitive computation of the systematic stiffness matrix. These shortcomings lead to the error accumulation and huge computational consumption, which encourage the reconsideration of force-based methods for elastoplastic problems. In this paper, a force-based Large Increment Method (LIM) is employed for the elastoplastic analysis of plates using a force-based 4-node quadrilateral plate element which is based on Mindlin–Reissner plate theory. The consistent elastoplastic flexibility matrix of plate element is derived and implemented to solve elastoplastic plate problems. Two numerical examples are presented to illustrate the mesh convergence of the plate element by solving the linear elastic thin and moderately thick plate problems by comparing with the analytical solutions and displacement-based plate elements. Two simple elastoplastic plate problems are presented to illustrate the accuracy and the computational efficiency of LIM by comparing with the results from the FEM software ABAQUS.

Development of the large increment method in analysis for thin and moderately thick plates

Many displacement-based quadrilateral plate elements based on Mindlin-Reissner plate theory have been proposed to analyze the thin and moderately thick plate problems. However, numerical inaccuracies of some elements appear since the presence of shear locking and spurious zero energy modes for thin plate problems. To overcome these shortcomings, we employ the large increment method (LIM) for the analyses of the plate bending problems, and propose a force-based 8-node quadrilateral plate (8NQP) element which is based on Mindlin-Reissner plate theory and has no extra spurious zero energy mode. Several benchmark plate bending problems are presented to illustrate the accuracy and convergence of the plate element by comparing with the analytical solutions and displacement-based plate elements. The results show that the 8-node plate element produces fast convergence and accurate stress distributions in both the moderately thick and thin plate bending problems. The plate element is insensitive to mesh distortion and it can avoid the shear locking for thin plate analysis.

The MLPG analyses of large deflections of magnetoelectroelastic plates

Abstract The von Karman plate theory of large deformations is applied to express the strains, which are then used in the constitutive equations for magnetoelectroelastic solids. The in-plane electric and magnetic fields can be ignored for plates. A quadratic variation of electric and magnetic potentials along the thickness direction of the plate is assumed. The number of unknown terms in the quadratic approximation is reduced, satisfying the Maxwell equations. Bending moments and shear forces are considered by the Reissner–Mindlin theory, and the original three-dimensional (3D) thick plate problem is reduced to a two-dimensional (2D) one. A meshless local Petrov–Galerkin (MLPG) method is applied to solve the governing equations derived based on the Reissner–Mindlin theory. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the centre of a circle surrounding it. The weak form on small subdomains with a Heaviside step function as the test function is applied to derive the local integral equations. After performing the spatial MLS approximation, a system of algebraic equations for certain nodal unknowns is obtained. Both stationary and time-harmonic loads are then analyzed numerically.

Analyses of functionally graded plates with a magnetoelectroelastic layer

A meshless local Petrov–Galerkin (MLPG) method is presented for the analysis of functionally graded material (FGM) plates with a sensor/actuator magnetoelectroelastic layer localized on the top surface of the plate. The Reissner–Mindlin shear deformation theory is applied to describe the plate bending problem. The expressions for the bending moment, shear force and normal force are obtained by integration through the FGM plate and magnetoelectric layer for the corresponding constitutive equations. Then, the original three-dimensional (3D) thick-plate problem is reduced to a two-dimensional (2D) problem. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the center of a circle surrounding the node. The weak-form on small subdomains with a Heaviside step function as the test function is applied to derive local integral equations. After performing the spatial MLS approximation, a system of ordinary differential equations of the second order for certain nodal unknowns is obtained. The derived ordinary differential equations are solved by the Houbolt finite-difference scheme. Pure mechanical loads or electromagnetic potentials are prescribed on the top of the layered plate. Both stationary and transient dynamic loads are analyzed.

Accurate variational approach for free vibration of variable thickness thin and thick plates with edges elastically restrained against translation and rotation

An efficient and accurate variational formulation is developed to study the vibration problem of variable thickness thin and thick plates with edges elastically restrained against both rotation and translation. The weak formulation with Ritz method is first employed to reduce the governing partial differential equations of motion of the plate to a system of ordinary differential equations. An analog procedure is then used to incorporate the natural boundary conditions. Although the proposed procedure requires some mathematical manipulations to derive the required boundary analog equations, it can produce lower upper bound solutions compared to the conventional Ritz method where the geometric boundary conditions can only be satisfied. The fast convergence and high accuracy of the proposed method are validated through convergence and comparison studies. Accurate solutions are achieved via few Ritz terms for all the cases considered.

Meshless Modelling of Laminate Mindlin Plates under Dynamic Loads

Collocation method and Galerkin method have been dominant in the existing meshless methods. A meshless local Petrov-Galerkin (MLPG) method is applied to solve laminate plate problems described by the Reissner-Mindlin theory for transient dynamic loads. The Reissner-Mindlin theory reduces the original three-dimensional (3-D) thick plate problem to a two-dimensional (2-D) problem. The bending moment and the shear force expressions are obtained by integration through the laminated plate for the considered constitutive equations in each lamina. The weak-form on small subdomains with a Heaviside step function as the test functions is applied to derive local integral equations. After performing the spatial MLS approximation, a system of ordinary differential equations of the second order for certain nodal unknowns is obtained. The derived ordinary differential equations are solved by the Houbolt finite-difference scheme as a time-stepping method.

Solving third- and fourth-order partial differential equations using GFDM: application to solve problems of plates

This paper describes the generalized finite difference method to solve second-order partial differential equation systems and fourth-order partial differential equations. This method is applied to solve the problem of thin and thick elastic plates.

Analytical Behavior Prediction for Skewed Thick Plates on Elastic Foundation

This paper presents analytical solutions for the problem of skewed thick plates under transverse load on a Winkler foundation, which has not been reported in the literature. The thick plate solution is obtained by using a framework of an oblique coordinate system. First, the governing differential equation in that system is derived, and the solution is obtained using deflection and rotation as derivatives of the potential function developed here. This method is applicable for arbitrary loading conditions, boundary conditions, and materials. The solution technique is applied to two illustrative application examples, and the results are compared with numerical solutions. The two approaches yielded results in good agreement.

A Thick Plate Problem Under the Action of a Body Force in Generalized Thermoelasticity

The two-dimensional problem for a thick plate is considered within the context of the theory of generalized thermoelasticity with one relaxation time under the action of a body force. The upper surface of the plate is subjected to a known temperature distribution, while its lower one is laid on a thermally insulated rigid foundation. Laplace and exponential Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. The inverse double transform is evaluated numerically. The distributions of the considered physical variables are obtained and represented graphically.